calculation of the moi

8/21/2019 Calculation of the MOI

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[ Up ] [ Moment of Inertia ] [ Calculation of the MOI ] [ Inertia Tensor ] [ Principal Axes ][ Transformation of the Inertia Tensor ] [ Angular Momentum ] [ Kinetic Energy ]

Calculation of the MOIPerpendicular-Axis TheoremParallel-Axis TheoremPhysical Pendulum & Direct MeasurementReferences and Related-Literature

Calculation of the MOI

In computation of the moment of inertia, one can replace the summation shown in [2] of Inertia

Tensor by an integration over the body:

[1]

where r = the perpendicular distance from the particle to the axis of rotation, and dm = the massof the particle which is a function of the density.

Thin Rod

Let's now apply [1] to a thin uniform rod shown in Figure 1. The MOI of the rod

about the Y axis is

Figur e 1

[2]

since

[3]

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where = the density of the rod, m = the mass of the rod, and L = the length of therod.

Circular Ring

The moment of inertia of the uniform circular ring shown in Figure 2 about the Z axis

(the symmetry axis) is

Figure 2

[4]

since

[5]

where dl = the length of the arc formed by d . [5] is also applicable to a circular cylinder.

Circular Disc

A uniform circular disc of radius R can be considered as a cascade of uniform

circular rings as shown in Figure 3. Thus, from [5], the moment of inertia about the Z axis (the symmetry axis) becomes


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Figure 3

[6]

since

[7]

[6] can be used in computing the MOI of a circular bar about its longitudinal axis aswell.

Sphere

A uniform sphere can be considered as a cascade of uniform circular discs asshown in Figure 4. From [6], the moment of inertia about the vertical axis (Z axis) is

Figure 4

[8]


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since

[9]

See BSP Equations for the moment-of-inertia equations for the geometric shapes commonlyused in the human body modeling.

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Perpendicular-Axis Theorem

Now, let's go back to the uniform circular disc (Figure 3). The moment of inertia of a uniformcircular disc about its perpendicular axis (Z axis) can be expressed as

[10]

since

[11]

In other words, the MOI about the Z axis is equal to sum of those about the X and Y axes. [10]is true for any rigid lamina: the MOI of any rigid lamina about an axis normal to the lamina planeis equal to the sum of the MOIs about any two perpendicular axes lying on the plane andpassing through the normal axis. This is the so-called perpendicular-axis theorem.

Since the circular disc has symmetric shape,

[12]

and, from [6]:

[13]

One can directly obtain [13] from [11] or

[14]

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[2] can be used in further developing [14] to obtain [13]. See BSP Equations for this approach.

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Parallel-Axis Theorem

Now, let's compute the MOI of a uniform circular column. Circular bar can be regarded as a

cascade of circular discs as shown in Figure 5. From [13]:

Figure 5

[15]

since

[16]

where mdisc = mass of the circular disc, and I y'y' (disc) = the MOI of the disc about the Y' axis.

[16] basically says that the MOI of a circular plate about the Y axis is equal to the sum of the

MOI about the parallel axis on the disc (Y' ) and the mass of the disc times square of thedistance between the two axes. This is the so-called parallel-axis theorem.

The MOI of the circular column is therefore

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[17]

since

[18]

Comparing [18] with [2], one can clearly see the difference in the MOI between a thin rod and athick rod (circular column). Similarly, [13] and [17] shows the difference in the MOI between acircular disc and a thick circular plate (circular column).

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Physical Pendulum & Direct Measurement

Unfortunately, the integration approach is possible only when the body has a known geometricshape. In the mathematical human body models such as Hanavan (1964) and Yeadon (1990),it is assumed that the body segments show a group of geometric shapes such as ellipsoid of revolution, elliptical solids, and stadium solids. See BSP Equations for the details.

If the body has a irregular shape, the integration approach has not much use and a directmeasurement must be attempted. Figure 6 shows a body with irregular shape which is rotatingfreely about an axis passing through its one end. The X axis is the axis of rotation, thus, thecenter of mass (CM) of the body moves within the YZ plane.

Figure 6

The torque produced by the weight of the body about the X axis is then

[19]

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where T x = the torque about the X axis, I xx = MOI of the body about the X axis, = angular

acceleration, m = the mass of the body, g = the gravitational acceleration (9.81 m/s2), and L =

the distance between the axis of rotation to the body's CM. For a small ,

[20]

and, from [19],

[21]

Solving [21] for , one obtains

[22]

where o = the amplitude, f = the frequency of the pendulum, = the phase angle, T = the

period of the pendulum. As shown in [22], the MOI of the body about the X axis, after all, can becomputed from the period of a small pendulum motion of the body. The MOI about the parallelaxis, which passes through the CM of the body, can be also computed based on the parallel-

axis theorem:

[23]

See Chandler et al. (1975) for an example of this approach.

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References and Related Literature

Chandler, R. F., Clauser, C. E., McConville, J. T., Reynolds, H. M. and Young, J. W. (1975)Investigation of inertial properties of the human body . AMRL-TR-74-137, AD-A016-485.DOT-HS-801-430. Aerospace Medical Research Laboratories, Wright-Patterson Air Force Base, Ohio.

Hanavan, E. P. (1964). A mathematical model of the human body . AMRL-TR-64-102, AD-608-463. Aerospace Medical Research Laboratories, Wright-Patterson Air Force Base, Ohio.

Yeadon, M. R. (1990). The simulation of aerial movement-II. A mathematical inertia model of the human body. J. Biomechanics 23, 67-74.

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calculation of the moi

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